LMFDB - L-function 2-1440-8.5-c3-0-14

Dirichlet series

L(s) = 1

− 5i·5^-s + 14.6·7^-s + 14.9i·11^-s + 85.6i·13^-s + 91.9·17^-s + 60.3i·19^-s + 1.33·23^-s − 25·25^-s − 25.5i·29^-s − 73.4·31^-s − 73.1i·35^-s − 211. i·37^-s − 330.·41^-s + 388. i·43^-s − 550.·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s + 0.789·7^-s + 0.411i·11^-s + 1.82i·13^-s + 1.31·17^-s + 0.728i·19^-s + 0.0121·23^-s − 0.200·25^-s − 0.163i·29^-s − 0.425·31^-s − 0.353i·35^-s − 0.938i·37^-s − 1.26·41^-s + 1.37i·43^-s − 1.70·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1440$ = $2^{5} \cdot 3^{2} \cdot 5$
Sign:	$-0.244 - 0.969i$
Analytic conductor:	$84.9627$
Root analytic conductor:	$9.21752$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1440} (721, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1440,\ (\ :3/2),\ -0.244 - 0.969i)$

Particular Values

$L(2)$	$\approx$	$1.713409354$
$L(\frac12)$	$\approx$	$1.713409354$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	5	$ 1 + 5iT $
good	7	$ 1 - 14.6T + 343T^{2} $
	11	$ 1 - 14.9iT - 1.33e3T^{2} $
	13	$ 1 - 85.6iT - 2.19e3T^{2} $
	17	$ 1 - 91.9T + 4.91e3T^{2} $
	19	$ 1 - 60.3iT - 6.85e3T^{2} $
	23	$ 1 - 1.33T + 1.21e4T^{2} $
	29	$ 1 + 25.5iT - 2.43e4T^{2} $
	31	$ 1 + 73.4T + 2.97e4T^{2} $
	37	$ 1 + 211. iT - 5.06e4T^{2} $
	41	$ 1 + 330.T + 6.89e4T^{2} $
	43	$ 1 - 388. iT - 7.95e4T^{2} $
	47	$ 1 + 550.T + 1.03e5T^{2} $
	53	$ 1 + 187. iT - 1.48e5T^{2} $
	59	$ 1 - 779. iT - 2.05e5T^{2} $
	61	$ 1 + 358. iT - 2.26e5T^{2} $
	67	$ 1 + 283. iT - 3.00e5T^{2} $
	71	$ 1 + 534.T + 3.57e5T^{2} $
	73	$ 1 - 1.01e3T + 3.89e5T^{2} $
	79	$ 1 + 1.11e3T + 4.93e5T^{2} $
	83	$ 1 - 1.19e3iT - 5.71e5T^{2} $
	89	$ 1 - 398.T + 7.04e5T^{2} $
	97	$ 1 - 278.T + 9.12e5T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.468239028040623252860867846275, −8.531709483391725114729286865465, −7.86509833039633263560878629117, −7.03955659854524342864683360339, −6.09299498113636368035458168741, −5.10627584891822800489269141177, −4.42713236875115850432220362335, −3.49249203344949001984907762668, −1.96695984620902953594708359900, −1.33369320010203856715521314652, 0.38356216023893664656053723338, 1.54454354255642101171967417945, 2.92881979772709855002474919323, 3.51301761950325449614398123804, 4.97360202884486482486324883181, 5.45895905901855330798538710349, 6.46456203773835254921559833377, 7.50625925900390481648097501079, 8.051058954702963058925086177154, 8.752936140969806218926954430064

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(1.713409354\)
\(L(\frac12)\)	\(\approx\)	\(1.713409354\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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